Is there a way of representing the minimax algorithm mathematically?
A simple google search gives plenty of results.

If you have a look at the entry in wikipedia for Minimax it has mathematical representations as well as some basic pseudocode and tree representations to help grasp the concept.

Proving it would be a matter of going through the regular methods of a mathematical proof and would probably be a bit complicated.  It is far easier to discuss this by way of graphs.